Theorem isacycgr1

Description: The property of being an acyclic graph. (Contributed by BTernaryTau, 11-Oct-2023)

	Ref	Expression
Assertion	isacycgr1	⊢ ( 𝐺 ∈ 𝑊 → ( 𝐺 ∈ AcyclicGraph ↔ ∀ 𝑓 ∀ 𝑝 ( 𝑓 ( Cycles ‘ 𝐺 ) 𝑝 → 𝑓 = ∅ ) ) )

Proof

Step	Hyp	Ref	Expression
1		fveq2	⊢ ( 𝑔 = 𝐺 → ( Cycles ‘ 𝑔 ) = ( Cycles ‘ 𝐺 ) )
2	1	breqd	⊢ ( 𝑔 = 𝐺 → ( 𝑓 ( Cycles ‘ 𝑔 ) 𝑝 ↔ 𝑓 ( Cycles ‘ 𝐺 ) 𝑝 ) )
3	2	imbi1d	⊢ ( 𝑔 = 𝐺 → ( ( 𝑓 ( Cycles ‘ 𝑔 ) 𝑝 → 𝑓 = ∅ ) ↔ ( 𝑓 ( Cycles ‘ 𝐺 ) 𝑝 → 𝑓 = ∅ ) ) )
4	3	2albidv	⊢ ( 𝑔 = 𝐺 → ( ∀ 𝑓 ∀ 𝑝 ( 𝑓 ( Cycles ‘ 𝑔 ) 𝑝 → 𝑓 = ∅ ) ↔ ∀ 𝑓 ∀ 𝑝 ( 𝑓 ( Cycles ‘ 𝐺 ) 𝑝 → 𝑓 = ∅ ) ) )
5		dfacycgr1	⊢ AcyclicGraph = { 𝑔 ∣ ∀ 𝑓 ∀ 𝑝 ( 𝑓 ( Cycles ‘ 𝑔 ) 𝑝 → 𝑓 = ∅ ) }
6	4 5	elab2g	⊢ ( 𝐺 ∈ 𝑊 → ( 𝐺 ∈ AcyclicGraph ↔ ∀ 𝑓 ∀ 𝑝 ( 𝑓 ( Cycles ‘ 𝐺 ) 𝑝 → 𝑓 = ∅ ) ) )